Solving Quadratic Equations Using Completing the Square

Environmental Science

The Basics of Quadratic Equations

If you're studying algebra, chances are you've come across quadratic equations. They're equations that include a variable that's squared, like this: ax^2 + bx + c = 0. Quadratic equations can be difficult to solve, but luckily, there are several methods you can use. One of them is called "completing the square." In this article, we'll explore how to use this method to solve quadratic equations.

Step-by-Step Process for Completing the Square

To complete the square of a quadratic equation, you need to follow a step-by-step process. Let's use the example equation: x^2 + 6x + 8 = 0, to guide us through the process.

Step 1: Write the Equation in Standard Form

The first step is to make sure your quadratic equation is written in standard form, which is ax^2 + bx + c = 0. In our example above, the equation is already in that form, so we can move on to step 2.

Step 2: Move the Constant to the Other Side of the Equation

In our example, 8 is the constant term. We're going to move it to the other side of the equation so that everything is on one side. Our equation becomes: x^2 + 6x = -8

Step 3: Divide the Coefficient of x by 2 and Square It

Now comes the tricky part. We need to take the coefficient of x (which is 6 in our example) and divide it by 2. That gives us 3. We then need to square that number, which gives us 9.

Step 4: Add the Number from step 3 to Both Sides of the Equation

We need to add 9 to both sides of the equation, like this: x^2 + 6x + 9 = -8 + 9 (x + 3)^2 = 1

Step 5: Take the Square Root of Both Sides of the Equation

We need to take the square root of both sides of the equation, which gives us: x + 3 = ±1

Step 6: Solve for x

Now, we need to solve for x. We'll start by subtracting 3 from both sides of the equation, like this: x = -3 ± 1 So, our two solutions are: x = -4 or x = -2

The Benefits of Completing the Square

There are a few benefits to using the completing the square method to solve quadratic equations. One is that it works for all quadratic equations, regardless of whether the coefficients are integers or fractions. It's also a good method to use when you need to find the maximum or minimum value of a quadratic equation, which we won't go into detail about in this article.

Conclusion

Completing the square is a useful method to know when solving quadratic equations. While it can be tricky to get the hang of at first, with practice, you'll be able to use it to solve any quadratic equation that comes your way.